The shape-based method can provide suitable initial guesses for trajectory optimization, which are useful for quickly converging\na more accurate trajectory. Combined with the optimal control theory, an optimized shape-based method using the finite Fourier\nseries is proposed in this paper. Taking the flight time-fixed case and the time-free case into account, respectively, the optimized\nshape-based method, which considers the first-order optimal necessary conditions, can guarantee that not only an orbit designed\nduring the preliminary phase is optimal, but also the thrust direction is not constrained to be tangential. Besides, the traditional\nshape-based method using the finite Fourier series, in which the thrust direction is constrained to be tangential, is developed for\nthe time-free case in this paper. The Earth-Mars case and the LEO-GEO case are used to verify the optimized shape-based method�s\nfeasibility for time-fixed and time-free continuous low-thrust trajectory design between circular coplanar orbits, respectively.The\noptimized shaped-based method can design a lower cost trajectory.
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